Question: Find one value of $x$ that is a solution to the equation: $(2x+3)^2-6x-9=0$ $x=$
Solution: We could solve for $x$ by expanding $(2x+3)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $-6x-9=-3({2x+3})$. This means that we can rewrite the equation as: $({2x+3})^2-3({2x+3})=0$ If we let ${p}={2x+3}$, we can see that this equation is in the form: ${p}^2-3{p}=0$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2-3{p}&=0\\\\ {p}({p}-3)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=3 \end{aligned}$ Since ${p}={2x+3}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${2x+3}=0\ \ \ \text{or} \ \ \ {2x+3}=3$ When we solve $2x+3=0$, we find that $x=-\dfrac{3}{2}$. When we solve $2x+3=3$, we find that $x=0$. In conclusion, the two solutions of the equation $(2x+3)^2-6x-9=0$ are $x=-\dfrac{3}{2}$ and $x=0$. [Is there another way to solve for x?]